For general $β\geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_β$ random point field. We prove that the increments of the limiting process are locally Brownian. When $β>1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{ö}lder for any $r<1-(1+β)^{-1}$. Furthermore for all $β\geq 1$ we show that the limiting process solves an SDE in a weak sense. When $β=2$ this limiting process is the Airy line ensemble.